1. Exponents and Exponential Functions
Chapter 8

2. 8-1 Zero and Negative Exponents
All nonzero numbers raised to the zero power = 1
e.g. 80 = 1, = 1, (-3422)0 = 1
A negative exponent does NOT make anything negative, it takes the entire power and moves it to the other half of a fraction
e.g and
Simplify each expression (write without negative exponents)
Ex1. Ex2. Ex3.

3. 8-2 Scientific Notation
Scientific notation is a common way to write very large and/or very small numbers
To write a number in scientific notation: write as the product of two factors in the form a x 10n where n is an integer and 1 < a < 10
The n represents the number of spaces the decimal point needs to move to return to its original place
If the original number is < 1, then n will be negative
If the original number is > 1, then n will be positive
Ex1. Are the numbers in scientific notation. If not, why?
A) x B) .83 x 10-4

4. Write each number in scientific notation
Ex2. 83,800,000,000 Ex
Numbers, as we are used to looking at them, are said to be in standard form
Ex4. Write 2.35 x 10-7 in standard form
Ex5. Write the numbers in order from least to greatest

5. 8-3 Multiplication Properties of Exponents
If you multiply nonzero powers with the SAME BASE, you add the exponents
e.g and
Simplify each expression, write without negative exponents
Ex Ex2.
To multiply two numbers in scientific notation
Multiply the coefficients
Multiply the powers of ten
Convert to scientific notation

6. Simplify each expression. Write each answer in scientific notation.
Ex Ex4.
Ex5. Complete the equation

7. 8-4 More Multiplication Properties of Exponents
When you raise a power to a power, multiply the exponents together
e.g and
Follow the order of operations if there are multiple steps
Simplify each expression.
Ex Ex2.
Ex Ex4.

8. Simplify each expression Ex5. Ex6. Ex7. Ex8.
If you raise a product to a power, raise each base to the power outside of the parentheses
e.g and
Simplify each expression
Ex Ex6.
Ex Ex8.

9. 8-5 Division Properties of Exponents
When you divide powers with the SAME base, subtract the exponents
e.g. and
Simplify each expression (no negative exponents)
Ex1. Ex2.
If you raise a quotient to a power, raise each base to the power outside of the parentheses
e.g and

10. Simplify each expression (no negative exponents)
Ex Ex4.
Ex5.

11. 8-6 Geometric Sequences
A sequence is geometric if you can multiply by the SAME number each time to get the next number
This number may be an integer, but it doesn’t have to be
The number you multiply by each time is called the common ratio
To find the common ratio, divide the 2nd number by the 1st number
Check this by dividing the 3rd number by the 2nd, etc.
A sequence is arithmetic if you can add the SAME number each time to get the next number (see section 5-6)

12. Formula for a geometric sequence
Ex1. 81, 27, 9, 3, …
A) find the common ratio
B) find the next two terms
Formula for a geometric sequence
n is the term position
a is the first term (some books use a1)
r is the common ratio
Ex2. A(n) = 3(-2)n-1
A) find the sixth term
B) find the twelfth term
Ex , 40, 8, …
A) find the next three terms
B) write a rule for the sequence

13. 8-7 Exponential Functions
Any function that is in the form y = a • bx where a is a nonzero constant, b > 0, b ≠ 1, and x is a real number is an exponential function
Ex1. Evaluate f(x) = 2 ∙ 3x for the domain {-4, 0, 3}
If |b|>1, then the graph is an exponential growth curve
If |b|<1, then the graph is an exponential decay curve
Exponential decay
Exponential growth

14. When graphing exponential curves, make a table of values and connect (at least 4 points)
Ex2. Suppose an investment of $2000 doubles in value every 15 years. How much is the investment worth after 45 years? Show your set up and answer.
Ex3. Suppose two mice live in a barn. If the number of mice quadruples every 3 months, how many mice will live in the barn after two years? Show your set up and answer.

15. 8-8 Exponential Growth and Decay
Both exponential growth and decay are in the form
It is growth if |b|>1
It is decay if |b|<1
The base b is the growth factor
The starting amount is a
When writing your equation, remember to define your variables first
When dealing with interest:
Add 100% to the interest rate and then change to a decimal
That is your growth factor (b)

16. Ex1. Suppose you deposited $800 in an account paying 3
Ex1. Suppose you deposited $800 in an account paying 3.4% interest compounded annually when you were born. Find the account balance after 18 years.
If the account is compounded more than once a year, it will change b and x
Divide the interest rate by the number of compoundings per year
Make sure the exponent reflects the number of times it is compounded total
Ex2. Suppose you deposit $800 in an account paying 3.4% interest compounded monthly when you were born. Find the account balance after 18 years.

17. If the initial amount is decaying, subtract the percent of decay from 100%, change it to a decimal, and then use it as the growth factor
Ex3. Suppose the population of a certain endangered species has decreased 2.4% each year. Suppose there were 60 of these animals in a given area in 1999.
A) Write an equation to model the number of animals in this species that remain alive in that area
B) Use your equation to find the approximate number of animals remaining in 2005.